Latus Rectum of the Hyperbola

We
will discuss about the latus rectum of the hyperbola along with the examples.

Definition of the latus rectum of the hyperbola:

The chord of the hyperbola through its one focus and perpendicular to the transverse axis (or parallel to the directrix) is called
the latus rectum of the hyperbola.

It is a double ordinate passing through the focus.
Suppose the equation of the hyperbola be \(\frac{x^{2}}{a^{2}}\) - \(\frac{y^{2}}{b^{2}}\) = 1 then, from the
above figure we observe that L\(_{1}\)SL\(_{2}\) is the latus rectum and L\(_{1}\)S is called the
semi-latus rectum. Again we see that M\(_{1}\)SM\(_{2}\) is also another latus rectum.

According to the diagram, the co-ordinates of the
end L\(_{1}\) of the latus
rectum L\(_{1}\)SL\(_{2}\) are (ae,
SL\(_{1}\)). As L\(_{1}\) lies on the hyperbola \(\frac{x^{2}}{a^{2}}\) - \(\frac{y^{2}}{b^{2}}\) = 1, therefore, we
get,